Discovering the Central Limit Theorem: Part Four (exponential distribution)

In this part we will sample from a population which has an exponential distribution. The graph of an exponential distribution is certainly not bell shaped. We will be looking (as we did in parts two and three) to see if the distribution of sample means is bell shaped.

[Maple Plot]

In our discussion we will need to use two facts about this exponential distribution (not covered in the text).

- The mean is 3.5

- The standard deviation is 6.25.

Click here to see the means of 500 samples of size 64 drawn from this exponential distribution 

statistics                  calculated values           
----------------------------------------------------------
count                          500                        
mean                           3.525910722                
median                         3.586728079                
mode                          no mode                     
range                          .883300464..5.817662087    
sample std dev                 .7826060502                
3rd quartile                   4.044336499                
1st quartile                   2.984703689

[Maple Plot]

We can make the four observations that we have seen in previous parts of this section.

Observation One

The mean of the sample means is close to the mean of the population. Here the mean of 3.525910722 is close to the population mean of 3.5.

Observation Two

The standard deviation of the sample means is smaller than the standard deviation of the population. Here .7826060502 is smaller than the population's standard deviation of 6.25.

Observation Three

The distribution of the sample means is bell shaped.

Observation Four

The population standard deviation divided by the square root of the sample size ( in this case the square root of 64) is very close to the standard deviation of the sample means. Here we get

[Maple Math] = [Maple Math] = .78125